Hirota Equation Research Articles

Rational Solutions to the Fourth Equation of the Nonlinear Schrödinger Hierarchy

This study concerns the research of rational solutions to the hierarchy of the nonlinear Schrödinger equation. In particular, we are interested in the equation of order 4. Rational solutions to the fourth equation of the NLS hierarchy are constructed and explicit expressions of these solutions are given for the first order. These solutions depend on multiple real parameters. We study the associated patterns of these solutions in the (x,t) plane according to the different values of their parameters. This work allows us to highlight the phenomenon of rogue waves, such as those seen in the case of lower-order equations such as the nonlinear Schrödinger equation, the mKdV equation, or the Hirota equation.

Double-pole anti-dark solitons for a Lakshmanan-Porsezian-Daniel equation in an optical fiber or a ferromagnetic spin chain

Under investigated in this paper is a Lakshmanan-Porsezian-Daniel equation that describes the nonlinear spin excitations in a (1+1)-dimensional isotropic biquadratic Heisenberg ferromagnetic spin chain with the octupole-dipole interaction or the propagation of the ultrashort pulses in a long-distance and high-speed optical fiber transmission system. Under certain parameter conditions, we simultaneously take the multi-pole phenomena and breather-to-soliton transitions into account, then utilize the second-order generalized Darboux transformation to derive the double-pole anti-dark solitons and graphically illustrate them. Asymptotic analysis is conducted to examine the interaction properties of double-pole anti-dark solitons, including their characteristic lines, amplitudes, phase shifts, slopes and position differences. Unlike the double-pole anti-dark solitons found in the Hirota equation, the ones in this study exhibit a distinct feature: Different soliton components share the same amplitude.

Simulations for the Schrödinger–Hirota equation arising in nonlinear optics in the presence of chromatic dispersion

Simulations for the Schrödinger–Hirota equation arising in nonlinear optics in the presence of chromatic dispersion

Riemann–Hilbert problem and soliton solutions with their asymptotic analysis for the focusing nonlocal Hirota equation with step-like initial data

The Riemann–Hilbert (RH) problem is developed to study the Cauchy problem of focusing space symmetric nonlocal Hirota (fsNH) equation with step-like initial data. Firstly, we analyze the characteristics of the Jost solutions and spectral functions, including analyticity, symmetry, and asymptotics as k→∞ and k→0. Then, we discuss three cases of zero point distributions of the spectral functions, and derive the expressions of spectral functions by constructing the special analytic functions and establishing the scalar RH problem. Furthermore, the constraint relations among the zeros of the spectral functions are analyzed. Next, the corresponding RH problem equipped with residual and singularity conditions is successfully derived. These singularity conditions are then transformed into a general residue condition, which reduces the problem of solving the RH problem with reflection-less potential to solving a system of algebraic equations. Ultimately, the corresponding soliton solutions of the fsNH equation and their asymptotics are derived from the solution of the RH problem.

The dynamic behaviors between double-hump solitons in birefringent fibers

In this paper, we research the fractional coupled Hirota equations with variable coefficients describing the collisions of two waves in deep oceans and the propagation of ultrashort light pulses in birefringent fibers and successfully acquire the double-hump one-soliton, two-solitons and N-solitons solutions via the Hirota bilinear method. At the same time, the Bäcklund transformation and the corresponding soliton solutions are also obtained. Based on the precise forms of the solitons solutions, we gain double-hump solitons images with different shapes including U-shape, V-shape and wave-type by assigning proper functions to the group velocity dispersion and the third-order dispersion and analyze the interaction dynamics of double-hump solitons. It is worth noting that the Hirota bilinear operators involved here are fractional rather than integer, which has never appeared in previous literatures.

On long-time asymptotic behavior and Painlevé asymptotic to the matrix Hirota equation

Abstract The nonlinear descent method is extended to study the long-time asymptotic behavior of the matrix Hirota equation with $4\times 4$ Lax pair in Schwartz space. The implementation of spectral analysis successfully transforms the Cauchy problem of the matrix Hirota equation into the corresponding high-order Riemann–Hilbert (RH) with $4\times 4$ jump matrix, and further analyses the established oscillation RH problem to study the asymptotic behavior of the solution in the space-time plane. Interestingly, the space-time plane $\{(x,t)|-\infty <x<+\infty , t>0\}$ can be divided into three different asymptotic regions based on the phase function and $\xi =x/t$. The first one is the oscillatory region $\xi <\frac{\alpha ^{2}}{3\beta }$, whose leading-term can be approximated applying the Weber equation with an error of $\mathcal{O}(t^{-1}\log t)$. The second region is the Painlevé region $\xi \approx \frac{\alpha ^{2}}{3\beta }$, whose leading-term can be approximated by the coupled Painlevé II equation, which is related to a $4\times 4$ matrix RH problem with an error of $\mathcal{O}(t^{-\frac{2}{3}})$. The last region is the fast decay region $\xi>\frac{\alpha ^{2}}{3\beta }$, which solution is rapidly decreasing as $t\rightarrow \infty $. Our results provide a detailed proof for the asymptotic analysis for the solution of the matrix Hirota equation on the complete space-time plane.

Dark solitons on elliptic function background for the defocusing Hirota equation

Abstract We investigate dark solitons lying on elliptic function background in the defocusing Hirota equation with third-order dispersion and self-steepening terms. By means of the modified squared wavefunction method, we obtain the Jacobi’s elliptic solution of the defocusing Hirota equation, and solve the related linear matrix eigenvalue problem on elliptic function background. The elliptic N-dark soliton solution in terms of theta functions is constructed by the Darboux transformation and limit technique. The asymptotic dynamical
behaviors for the elliptic N-dark soliton solution as t → ±∞ are studied. Through numerical plots of the elliptic one-, two- and three-dark solitons, the amplification effect on the velocity of elliptic dark solitons, and the compression effect on the soliton spatiotemporal distributions produced by the third-order dispersion and self-steepening terms are discussed.

On a Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation with step-like initial data

On a Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation with step-like initial data

Non-homogeneous initial boundary value problems and small data scattering of the Hirota equations posed on the half line

Non-homogeneous initial boundary value problems and small data scattering of the Hirota equations posed on the half line

Optical soliton solutions and their nonlinear dynamics described by a novel time-varying spectral Hirota equation with variable coefficients

Optical soliton solutions and their nonlinear dynamics described by a novel time-varying spectral Hirota equation with variable coefficients

Super-regular breathers induced by the higher-order effects in coupled Hirota equations

Super-regular breathers induced by the higher-order effects in coupled Hirota equations

Multi-pole soliton of discrete integrable equations and modified Riemann-Hilbert approach: discrete Hirota equation

In this study, the Riemann-Hilbert approach was developed and applied to the discrete Hirota equation. We constructed a modified Riemann-Hilbert problem compatible with the discrete Hirota equation and derived a reconstruction formula for its solutions. Because the characteristic function contains a potential, we modify the Riemann-Hilbert approach to make the Riemann-Hilbert matrix have good asymptotic properties. We believe that the modified Riemann-Hilbert approach can also be applied to other discrete integrable models. By using the direct method of Laurent series, we obtained the expression of multi-pole solutions for the discrete Hirota equation and demonstrated the dynamic behavior of some solutions.

Abundant optical solutions for the stochastic Schrödinger–Hirota equation in quantum mechanics, science and engineering

Abundant optical solutions for the stochastic Schrödinger–Hirota equation in quantum mechanics, science and engineering

Riemann‐Hilbert approach for the integrable discrete Hirota equation with bounded boundary conditions in the presence of a discrete spectrum

Riemann‐Hilbert approach for the integrable discrete Hirota equation with bounded boundary conditions in the presence of a discrete spectrum

Mixed localized waves and their interaction structures for a spatial discrete Hirota equation

Mixed localized wave solutions and interactions are of great significance in nonlinear physical systems. This paper aims to investigating the generalized (m,N-m)-fold Darboux transformation and the mixed localized wave solutions of a spatial discrete Hirota equation. First, we construct the generalized (m,N-m)-fold Darboux transformation for the spatial discrete Hirota equation, which can produce the interactions between the breathers, degenerate breathers and rogue waves. For the Darboux transformation formula, we discuss the above order-1,2,3 localized wave solutions, as well as their dynamics by choosing the number of m = 1. We plot some specific examples such as the spatial (time)-periodic breather, second-order and third-order degenerate breathers and higher-order rogue waves with novel patterns. Furthermore, when m > 1, we give several kinds of mixed interaction solutions between the first-order rogue waves and first (second)-order (degenerate) breathers, between the first-order breather and second-order degenerate breathers, between second-order rogue waves and first-order breathers. At last, we also sum up the various mathematical features of the degenerate breathers and the mixed localized wave solutions.

Transient modes for the coupled modified Korteweg-de Vries equations with negative cubic nonlinearity: Stability and applications of breathers.

Dynamics and properties of breathers for the modified Korteweg-de Vries equations with negative cubic nonlinearities are studied. While breathers and rogue waves are absent in a single component waveguide for the negative nonlinearity case, coupling can induce regimes of modulation instabilities. Such instabilities are correlated with the existence of rogue waves and breathers. Similar scenarios have been demonstrated previously for coupled systems of nonlinear Schrödinger and Hirota equations. Both real- and complex-valued modified Korteweg-de Vries equations will be treated, which are applicable to stratified fluids and optical waveguides, respectively. One special family of breathers for coupled, complex-valued equations is derived analytically. Robustness and stability of breathers are studied computationally. Knowledge of the growth rates of modulation instability of plane waves provides an instructive prelude on the robustness of breathers to deterministic perturbations. A theoretical formulation of the linear instability of breathers will involve differential equations with periodic coefficient, i.e., a Floquet analysis. Breathers associated with larger eigenvalues of the monodromy matrix tend to suffer greater instability and increased tendency of distortion. Predictions based on modulation instability and Floquet analysis show excellent agreements. The same trend is obtained for simulations conducted with random noise disturbances. Linear approaches like modulation instabilities and Floquet analysis, thus, generate a very illuminating picture of the nonlinear dynamics.

A KP-mKP hierarchy via pseudo-differential operators with two derivations

By using pseudo-differential operators containing two derivations, we extend the Kadomtsev–Petviashvili (KP) hierarchy to a certain KP-mKP hierarchy. For the KP-mKP hierarchy, we derive its Bäcklund transformations, bilinear equations of Baker–Akhiezer functions and Hirota equations of tau functions. Moreover, we show that this hierarchy is equivalent to a subhierarchy of the dispersive Whitham hierarchy associated to the Riemann sphere with its infinity point and one movable point marked.

The coupled Hirota equations with a 3×3$3\times 3$ Lax pair: Painlevé‐type asymptotics in transition zone

AbstractWe consider the Painlevé asymptotics for a solution of the integrable coupled Hirota equations with a Lax pair whose initial data decay rapidly at infinity. Using the Riemann–Hilbert (RH) techniques and Deift–Zhou nonlinear steepest descent arguments, in a transition zone defined by , where is a constant, it turns out that the leading‐order term to the solution can be expressed in terms of the solution of a coupled Painlevé II equations, which are associated with a matrix RH problem and appear in a variety of random matrix models.

Data-driven solutions and parameter discovery of the extended higher-order nonlinear Schrödinger equation in optical fibers

In this paper, we investigate an extended higher-order nonlinear Schrödinger equation that includes third-order and fourth-order dispersion terms. By considering higher-order effects, the equation can more accurately simulate and predict the propagation behavior of ultrashort pulses in optical fibers, e.g., pulse splitting, compression and stability. In the optical fiber sensors, the sensitivity and resolution of the sensor can be improved by utilizing nonlinear effects. The analysis of the extended higher-order nonlinear Schrödinger equation helps to understand how nonlinear effects affect the sensing signal, thereby designing more accurate sensors. Specific coefficients of each term in the equation are studied for different scenarios, such as optical fiber design and signal propagation in optical communications. Under certain conditions, the extended higher-order nonlinear Schrödinger equation can be degenerated into the Hirota equation or the Lakshmanan–Porsezian–Daniel equation. With the development of numerical algorithms, the utilization of deep neural networks for solving equations has become increasingly attractive. With given initial conditions and boundary conditions, the physical information is embedded into the neural network in the form of loss functions. We efficiently derive numerical solutions to the extended equation for one-soliton, two-soliton and rogue-wave cases by training weights and biases in physics-informed neural networks. Compared with traditional numerical methods, physics-informed neural networks yield lower prediction errors with less data volume, thus will reduce the cost and time consumptions for data sampling in practical engineering. Furthermore, we solve the inverse problem associated with the extended equation. This approach is critical for extracting fundamental parameters of optical systems from observable data, thereby deepening our understanding of complex optical behavior and helping to improve the performance of optical fiber communication technologies.

Novel multi breather like, periodic, hybrid periodic and singular periodic waves of the Schrödinger–Hirota equation having the parabolic-law nonlinearity

Novel multi breather like, periodic, hybrid periodic and singular periodic waves of the Schrödinger–Hirota equation having the parabolic-law nonlinearity